Matrices & Calculus|MA3151 Arrear Examination important questions|Engineering maths.

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Practice questions: Unit 1: Part-B 1. Reduce the quadratic form 6x ^ 2 + 3y ^ 2 + 3z ^ 2 - 4xy - 2yz + 4xz into canonical Form by an orthogonal reduction. 2. Reduce the quadratic form x ^ 2 + y ^ 2 + z ^ 2 - 2xy - 2yz - 2zx to canonical form through an orthogonal transformation. 3. Reduce the quadratic form 3x ^ 2 + 5y ^ 2 + 3z ^ 2 - 2yz + 2zx - 2xy to the canonical form through an orthogonal transformation. 4. Reduce the quadratic form Q = 2x_{1}*x_{2} + 2x_{2}*x_{3} + 2x_{3}*x_{1} into canonical Form. 5. Reduce the quadratic form x ^ 2 + 5y ^ 2 + z ^ 2 + 2xy + 6zx + 2yz to a canonical form through an orthogonal transformation. Unit-2: 1. Find the absolute maximum and absolute minimum values of the function f(x) = 3x ^ 4 - 4x ^ 3 - 12x ^ 2 + 1 on the interval [-2,3]. Unit-3: Part-B 1. Find the Taylor's series expansion of f(x, y) = x ^ 2 * y ^ 2 + 2x ^ 2 * y + 3x * y ^ 2 in power of (x + 2) and (y-1) up to the second degree term. 2. Find the Taylor's series expansion of x ^ 2 * y ^ 2 + 2x ^ 2 * y + 3x * y ^ 2 powers of (x+2) and (y - 1) up to 3 ^ (rd) degree term. 3. Expand x ^ 2 * y + 3y - 2 in powers of (x - 1) and (y + 2) using Taylor's expansion. 4. Find the Taylor's series expansion of e ^ x * sin y in term of x and y up to third degree 5. Expand e* cosy about (0, pi/2) up t to the third term using Taylor' s series 6. Expand e ^ x * log(l + y) in powers of x and y up to terms of third degree. 7. A rectangular box open at the top, is to have volume of 32cc. Find the dimension of the box, that requires the least material for its construction. Unit-4: Part-B 1. Evaluate e sin dxdx, by using integration by parts. 2. Evaluate ecos dxdx by using integration by parts. 3. Establish a reduction formula for I_{n} = integrate sin^n x dx Hence find integrate sin^n x dx from 0 to pi/2 4. Establish a reduction formula for I_{n} = integrate cos^n x dx Hence find integrate cos^n x dx from 0 to pi/4 Unit-5: Part-B 1. Find the volume of the tetrahedron bounded by the coordinate planes and the plane x/a + y/b + z/c = 1 2. Find the volume of that portion of the ellipsoid using triple integration. (x ^ 2)/(a ^ 2) + (y ^ 2)/(b ^ 2) + (z ^ 2)/(c ^ 2) = 1 which lies in the first octant 3. Find the volume of the region bounded by the parabolic z = x ^ 2 + y ^ 2 and the plane z = 4 . 4. Find the volume bounded by cylinder x ^ 2 + y ^ 2 = 4 and the planes y + z = 4 and z = 0 using triple integral. 5. Find the value of the xyz dxdydz through the positive spherical octant for which x ^ 2 + y ^ 2 + z ^ 2

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